Question

A barrels manufacturer can produce up to 300 barrels per day. The profit made from the sale of

these barrels can be represented by the polynomial p(x) = -x² + 350x – 6600. He needs to pay his
employees on a daily basis irrespective of the number of barrels sold. If x is the number of barrels
sold then based on this information answer the following questions:
(i) What is the type of the above polynomial? (1)
(ii) What will be his loss in rupees if he is not able to sell any barrels? (1)
(iii) How many minimum barrels does he need to manufacture so that there is no profit or no
loss for him ?

Answer 1

Explanation:

(i) The type of the given polynomial is a quadratic polynomial. It is a second-degree polynomial because the highest exponent of the variable x is 2.

(ii) To find the loss if no barrels are sold, we substitute x = 0 into the polynomial p(x):

p(0) = -(0)² + 350(0) - 6600

p(0) = -6600

Therefore, if no barrels are sold, the loss would be 6600 rupees.

(iii) To determine the minimum number of barrels required to have no profit or no loss, we need to find the x-value where the profit function p(x) equals zero.

p(x) = 0

Solving the quadratic equation:

-x² + 350x - 6600 = 0

We can factorize the equation:

-(x - 20)(x - 330) = 0

Setting each factor equal to zero:

x - 20 = 0 -> x = 20

x - 330 = 0 -> x = 330

So, the minimum number of barrels required to have no profit or no loss is either 20 barrels or 330 barrels.